Combining weight, balance and swingweight

The relation between weight (mass) and force is something most people have a natural feel for: You have to apply more force to accelerate a heavy ball than a light one when throwing it. The formula is also straight forward:
F=m*acc

When you move in a curve, as when swinging a tennis racquet, you have something called swingweight that is supposed to tell you how it feels. The problem, however, is that most people don't know what swingweight really is. And even worse, the values you get from manufacturers and resellers are only valid when you rotate the racquet around a point 10 cm up the handle, a type of swing that rarely occurs in tennis.

I will try to shed some light on this and propose a curve where you can compare different racquets for different types of swings. Say that we have a racquet where we apply a force F at the handle and swing it around a point p. The the swing radius r says if the swing is short or long. When swinging we are interested in accelerating the racquet head:

If we look at the relation between F and the acceleration of the head acc we can define an equivalent mass me that tells us how much force you have to apply to get a certain acceleration, i.e. how heavy the racquets feels for different kinds of swings:
me=F/acc

We can then plot me and compare different racquets. But instead of plotting me against r I will plot it against 15/r. I that way we will get a convenient scale where 0 means moving it without rotation (a block) and 1 means whipping it around the wrist (5 cm outside the handle).

As you can see the Pro Staff is heavier for long swings (as expected), but it is also heavier for shorter swings despite that Cierzo has a higher swingweight. The reason is that the swingweight doesn't the the full story, even for a short swing. The diagram therefore gives you a way to compare these two racquets for different swings.

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For those who want hear some more details we need to define some lengths:

You can then calculate me in terms of the swing radius r:

Where m is the weight and sw is the swingweight of the racquet. I have used d=40 cm (i.e. 50 cm from the but) in the diagram above.

For those who want to play around with the figures I have an Excel-sheet that you can download here
Edit: There is an alternative and better version with a new excel sheet presented in post 40

Is there an ideal personal curve, you think, which will give a nice feel of the forces needed in the stroke.

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That is a very good question!
No, I think that would be to push the significance of the curve a little to far. Different people will still like different racquet behavior. You will also vary the radius during the swing. But maybe you could take a racquet you like and use the curve as the basis when buying or customizing other racquets.

I would use the curve in the first post to say that the Pro Staff 90 is "always heavier, especially for long swings" something that is not obvious when you look at the weight, swingweight and balance.

I am having trouble filling in the formula. Could you provide an example of the calculations?

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The excel sheet (see the first post) has two racquets included as an example.

Or do you mean how I derived the formula? I didn't include that since I didn't want to scare people with to much math to begin with. But it is coming, I just have to type it out so it looks nicer.

And I don't get the 15/r part.

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It is only there to rescale the curve to a more convenient form. It doesn't change the curve.

r is infinite when you move the racquet without rotation, but that is not so easy to show in a curve. 1/r = 0 when r is infinite, so thats easier to plot .

At the other end I picked 15 cm as a reasonable shortest swing radius. Since r is measured from 10 cm up the handle where the force is applied, r=15 means 5 cm outside the end but. Which is somewhere around the wrist. So when r=15 then 15/r=1. You can of course use another point as reference, r=10 or r=20 if you like, as long as you don't go all the way to r=0 when the calculation becomes irrelevant.

I don't mean how you got the formula figured out, but I am having trouble filling in the numbers and then get the correct me. I think I do it in the wrong units.
Let's say I have a racket with the following specs: 335g, 32cm balance, sw 330. Can you show me the math to calculate the me?

I don't mean how you got the formula figured out, but I am having trouble filling in the numbers and then get the correct me. I think I do it in the wrong units.
Let's say I have a racket with the following specs: 335g, 32cm balance, sw 330. Can you show me the math to calculate the me?

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Your units looks fine.

In the Excel sheet the length c to the balance point (cg) is calculated from the HH/HL value. But if you want to enter the balance directly just type in your value -10 at c (in your case 22 cm). That will destroy the function that calculates it from HH/HL.

I don't know the length of your racquet, but assuming it is 69 cm the curve will look like this (using the new version of the spreadsheet):

I have also read Travlerajm's formulas and tried them. I don't know if you have heard of Mgr/I? If so, can this formula interact with yours?

It seems difficult, because according to Mgr/I, if I add weight at the top of the rackethandle, the relative speed of rackethead becomes higher relative to the hand. My virtual racket has a lower me at every point and the only outstanding difference between my racket and the prostaff in the example is that the prostaff has more weight. more static weight gives a higher Mgr/I, which should make it easier on the wrist. But then, according to your formula, the prostaff has more me at the wrist. Which should feel heavier then, I think. I am not questioning your formula, I just want to know if I may be comparing two different things.

On a side note. Once upon a time I tried adding weight on the top of the handle of a K Six-One 95 to speed up the racket head, but I never got it right. It kept feeling too heavy. Probably to much me.

I have experimented with your formula and I think it gives very valuable information. I am still experimenting with what spec change changes the curve in which way.

Could you share what combination of spec changes, makes the curve flatter/steeper, more curved or less curved?

It a challenge for me to figure out if you like a certain racket and thus the corresponding curve, if it acts like a blueprint.
Say you want the same feel during the stroke, but with just more me. Should the curve just being lifted upward or does the curve has to change a little to get the same feel? The trouble a lot players have, is that they like a certain racket, but want more heft. When adding weight it's tough to keep the same feel with a heavier racket.

I have experimented with your formula and I think it gives very valuable information. I am still experimenting with what spec change changes the curve in which way.

Could you share what combination of spec changes, makes the curve flatter/steeper, more curved or less curved?

It a challenge for me to figure out if you like a certain racket and thus the corresponding curve, if it acts like a blueprint.
Say you want the same feel during the stroke, but with just more me. Should the curve just being lifted upward or does the curve has to change a little to get the same feel? The trouble a lot players have, is that they like a certain racket, but want more heft. When adding weight it's tough to keep the same feel with a heavier racket.

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I started this thread to get some input on how use the curve, so your view is a s good as mine

I would guess that a racquet that follows a similar curve but on a lower level (like your racquet compared to the Pro Staff) should feel similar, but a little lighter in all situations. A heavy, head light, low swingweight racquet will get a flatter curve than a more top heavy. What the curve can show is at what types of swings it actually is heavier, it might only be for very long swings, something that not is obvious when you just look at the basic data.

Let me give two examples: I have started from rather generic racquet, 70 cm long, weight 300 g, swingweight 320, even balance. I have then added 20 g of weight, in one case at the top, in the other at the handle. The latter only differs from the original at long swings and the curves quickly merge. The top heavy racquet is heavier all the way and difference increases for short swings:

In the other example I have also added 20 g, but in this case either all in the middle or half at the top and half at the handle. Putting everything in the middle increases the weight evenly, whereas the more "polarized" racquet increases a little more for short swings. But the difference is fairly small I would say.

Why not just use the parallel axis theorem to find the effective swingweights at the wrist, elbow, shoulder, etc.?

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A good question. And the answer is that it is exactly what I am doing
The numerator in the equation in post 1 is Moment of Inertia around p

But choose to transform it into a weight for a couple of reasons:
- I think that weight is easier to understand
- The equivalent weight nicely converges to the weight of the racquet for very long swings, i.e. at the y-axis. Whereas the swingweight has no meaning when the racquet doesn't rotate.
- Weight gives the relation between force and acceleration whereas swingweight gives the relation between Moment and angular acceleration, again less intuitive.

I have this idea; I'll set-up 2 rackets, which I will try to get to feel the same, except that one is heavier overall. IOW. I want to achieve the same feel with both during play, but one will just feel heavier. I will try this with this method and if I succeed I will then carefully measure the specs.

I hope that you could make another plot of those two rackets. Then we can compare the different curves.

I know it's a personal experiment, but maybe we will learn something of it.

I have this idea; I'll set-up 2 rackets, which I will try to get to feel the same, except that one is heavier overall. IOW. I want to achieve the same feel with both during play, but one will just feel heavier. I will try this with this method and if I succeed I will then carefully measure the specs.

I hope that you could make another plot of those two rackets. Then we can compare the different curves.

I know it's a personal experiment, but maybe we will learn something of it.

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It will be interesting to hear, just send me the data and I will plot them.

JohnB, just curious, did you focus mostly on getting the forehand to feel the same, or were you also thinking about backhand?

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Mostly forehand, but backhands (one handed) also felt about the same. When I was satisfied with similar feel I switched between the rackets and it took me virtually no time to adjust. I just played and felt very in tuned with both of them.

Mostly forehand, but backhands (one handed) also felt about the same. When I was satisfied with similar feel I switched between the rackets and it took me virtually no time to adjust. I just played and felt very in tuned with both of them.

Also with volleys, smashes etc.

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Interesting. The reason I ask is that based on the specs you provided, the MgR/I values of your two rackets are almost exactly the same (21.20 and 21.26). The MgR'/I' values (for two-handed backhand) are much farther apart (23.01 and 23.31), but since you have a one-handed backhand, you wouldn't have noticed. I'm very curious what Sten's curves end up looking like. (Sten, I apologize for bringing MgR/I into this thread - just thought it was an interesting observation. I'm excited to see how useful your equation/curves can be.)

Interesting. The reason I ask is that based on the specs you provided, the MgR/I values of your two rackets are almost exactly the same (21.20 and 21.26). The MgR'/I' values (for two-handed backhand) are much farther apart (23.01 and 23.31), but since you have a one-handed backhand, you wouldn't have noticed. I'm very curious what Sten's curves end up looking like. (Sten, I apologize for bringing MgR/I into this thread - just thought it was an interesting observation. I'm excited to see how useful your equation/curves can be.)

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Don't apologize, it deals with the same stuff (swingweight, mass etc) as the curves, so it is relevant to discuss it here. I think that there might be some interesting stuff buried in there somewhere and that is why I spent time discussing it. The problem is that travlerajm treats it as some kind of holy grail and applies it everywhere. Like the unfounded statement that you should use MgR/I for one handed swings and MgR'/I' for two handed (sorry, no critique of you), not to speak of the hilarious theory that you need to include g since gravity varies in different places in the world.

As an example of the connection I took the two Wilson racquets I used in the first post and modified them to MgR/I = 21.0. I think that they are interesting since they are very different to begin with. The Pro Staff has a MgR/I = 21.5 which is fairly close to the magical number so it was enough to add 8 g to the top leading to m=365 g, sw=354, balance=22.6 cm. The Cierzo Two has MgR/I = 19.1 and to bring it up to 21.0 the least I could get away with was adding 85 g at 25 cm. It resulted in a racquet with m=363 g, sw=369, balance=34.3 cm (1 pt HL). And the resulting curves:

Giving them equal MgR/I certainly brought the curves closer together. On the other hand the original racquets are close together for short swings despite having radically different MgR/I.

If the shape of the curve is a personal preference, regardless how high or low it is situated (within reason) I have a few thoughts about it:

If one increases MgR/I by putting mass on top of the handle, the curve remains the same, but situated higher, thus the racket will feel the same, only heavier, despite that the rackethead should travel faster relative to the handle.

If one decreases or increases MgR/I by subtracting or adding swingweight, the curve will flatten or steepen, thus the rackethead will travel faster or slower and feel different.

Is this what you mean Sten, that MgR/I is a consequence of weightdistribution and not the cause of what makes a racket play sweet?

No. ω is angular acceleration (as said in the document). You can also see that I have used M=ω*I which indicates that it is angular acceleration. So the formula is correct.

However, ω is often used to denote angular velocity so you are right that it is confusing. I will change in the document. Thanks for being a careful reader.

/Sten

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According to all textbook about angular motion, Ѡ is angular velocity. So formula 4 was wrong.
Right now you start using α – angular acceleration without proper corrections and all your formulas become wrong.

According to all textbook about angular motion, Ѡ is angular velocity. So formula 4 was wrong.
Right now you start using α – angular acceleration without proper corrections and all your formulas become wrong.

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ω isn't angular velocity, but it is often used to denote (stand for) angular velocity. But you can use any symbol you like as long as you are consistent (and I was). I have changed so now I am using the more common α to denote angular acceleration to avoid confusion. While at it I also changed the Moment (torque) from M to the more common τ in the latest version.

There are no standards saying what symbol you must use for a certain variable, just common practice. I agree that the common practice helps avoiding confusion, but it doesn't change the equations. And the common practice has changed since I taught at the university

If the shape of the curve is a personal preference, regardless how high or low it is situated (within reason) I have a few thoughts about it:

If one increases MgR/I by putting mass on top of the handle, the curve remains the same, but situated higher, thus the racket will feel the same, only heavier, despite that the rackethead should travel faster relative to the handle.

If one decreases or increases MgR/I by subtracting or adding swingweight, the curve will flatten or steepen, thus the rackethead will travel faster or slower and feel different.

Is this what you mean Sten, that MgR/I is a consequence of weightdistribution and not the cause of what makes a racket play sweet?

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The swing radius tells you the relation between racquet head and handle speed. At the left side of the curve they are the same, at the right side the handle speed is close to zero. So if you pick a point say 15/r = 0.5, r is 30 cm and the acc point moves (30+40)/30 = 2.3 times faster than the handle. And that is the same for all curves at that point. (I am not sure if that answers your question).

And yes, MgR/I is a measure of weight distribution. It is taking one such measure (MR) and dividing it with another (I). It also means that a very heavy racquet (100 kg) and a very light racquet (10 g) with similar weight distribution can have the same MgR/I.

A fun exercise in connection with this is to assume that the racquet has a completely even weight distribution. MgR/I=21 will then lead to the result that the ideal racquet is 70 cm long, but of arbitrary weight!

The swing radius tells you the relation between racquet head and handle speed. At the left side of the curve they are the same, at the right side the handle speed is close to zero. So if you pick a point say 15/r = 0.5, r is 30 cm and the acc point moves (30+40)/30 = 2.3 times faster than the handle. And that is the same for all curves at that point. (I am not sure if that answers your question).

And yes, MgR/I is a measure of weight distribution. It is taking one such measure (MR) and dividing it with another (I). It also means that a very heavy racquet (100 kg) and a very light racquet (10 g) with similar weight distribution can have the same MgR/I.

A fun exercise in connection with this is to assume that the racquet has a completely even weight distribution. MgR/I=21 will then lead to the result that the ideal racquet is 70 cm long, but of arbitrary weight!

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Yes I understand.

This method/formula is truly amazing. I have set-up 5 rackets differently (different specs) and they are all very playable.

Sten, I have also tested (before this thread) the influence of mass locations, instead of just the specs. I went on the court with two rackets with the same specs, as close as I could, but with a different weightdistibution. They played quite differently. One was hitting flatter than the other. I don't think this is covered in the formula, or is it?

If it isn't, you can determine your favorite specs. Then finetune the playingcharacteristics by adjusting the weightdistribution within the specs you like.

Could you map Travlerjm racquet which is 383grams with 31.877cm balance and 365 Swingweight and then fine the same curve but in a lighter racquet that is only 320grams? Can you tell me what the balance and swingweight needs to be?

I have also tested (before this thread) the influence of mass locations, instead of just the specs. I went on the court with two rackets with the same specs, as close as I could, but with a different weightdistibution. They played quite differently. One was hitting flatter than the other. I don't think this is covered in the formula, or is it?

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It is included. If you look at the formula:

You can see that both the swing weight (sw) and the balance (c) are in the numerator. So increasing either will increase me and make the curve steeper, but in a slightly different way.

This spec is so polarized (and lower powered than any players frame on the market) it might be nearly impossible to achieve from the specs of 99% of retail frames. It 'might' be doable with a prostock frame, with all the lead at 12 and in the butt, and making sure there is a minimum amount of weight added in between. (ie. no leather grip, no dampener, light strings, etc.)

edit: just tried it on the customization tool for the Ti.radical mp (a very customizable retail frame) and the closest you can get for 320g is around 32.8 cm balance with 327.5 sw (which will give you a much more powerful frame).

This spec is so polarized (and lower powered than any players frame on the market) it might be nearly impossible to achieve from the specs of 99% of retail frames. It 'might' be doable with a prostock frame, with all the lead at 12 and in the butt, and making sure there is a minimum amount of weight added in between. (ie. no leather grip, no dampener, light strings, etc.)

edit: just tried it on the customization tool for the Ti.radical mp (a very customizable retail frame) and the closest you can get for 320g is around 32.8 cm balance with 327.5 sw (which will give you a much more powerful frame).

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Would you like to try the following experiment? I would love to know about your opinion/feel on the court if you would play with your current frame and then compare it with a frame customized as if weight is removed from the middle. IOW play with the same, but lower curve and compare whether the lower MgR/I is as noticeable as you would have anticipated if the curve wouldn't have had the same shape.

Now I'm remembering I have tried similar experiments in the past. The result is an extremely spin friendly setup (extreme loopy groundstrokes, great arcing topspin serves), and a flexy feel to the impact. The racquet needed a lot of wrist added before contact on each stroke because the racquet head lagged behind so much (a polarized, disconnected feel between the tip and the handle), like a barbell. It was difficult to hit accurate flat shots, volleys were very grabby and hard to pull off, and I had to prepare earlier for overheads.

Just when you though you understood it, I am going to hit you with an alternative curve

Well, actually it is showing the same thing and I think it is an improvement. Instead of showing the equivalent mass, I am showing the equivalent mass for the racquet divided by the equivalent mass for a reference frame. And as the reference I am using a 70 cm long 320 g heavy* racquet with a completely even mass distribution and balance. By doing this it will be easier to see the difference between racquets. And the slope of the curve now also means something: A rising slope means that it is relatively heavier to swing at a short radius compared to the neutral racquet. And inverse for the falling curve.

Three examples, first our old friends the Pro Staff and the Cierzo:

Then Johns two racquets:

Finally an answer to DEH's question of taking travlerajm's racquet and make it 320 g. With balance 32 cm and swingweight 305 they look similar:

*Note. I picked the weight of the reference racquet to be 320 g as it is an average of the racquets in the tenniswarehouse database. The value, however, is not so important since it will only shift the scale of the y-axis up or down.

Thanks for posting all the good info. Here is what I got. I have tune my daughter's racquet (she is 15) for a long time using Travlerajm's method and it seem to work very well. I ended up with 324g with a balance of 32.7 and swingweight of 310. She was using a windsheild style forehand and it seemed that the shots where very spinny but not a lot of power. I tried all kinds of strings and tensions and I could not get any more power out of it. So we changed here swinging style to a Djokovic style swing which took about a month to do. Now that racquet will not work at all. The balls fly all over the place. No control at all. Tried more strings and no luck. So I tried another racquet. It was an nTour Wilson and it has some weird specs so it could be fake I don't know. Well she started hit a little better. So I just started adding lead to the top of the racquet and it got better and better and better. I added 18 grams at 12. So now her shots are controlled with a lot of spin and power. The specs are 314g with a balance of 34 and 337 swingweight. Not really close to the other specs. So I was just trying to figure out what is so different between the two styles of hitting and the curve on the racquet map. Here is a link for some WTA specs.http://www.hdtennis.com/grs/pro_racquet_specs/201207mercury-insurance-open.html
Look at Sloane Stevens racquet. Thanks again.

What puzzles me is that the lighter Travlerajm racket has virtually the same MgR/I and the same shape of the curve. But if you add weight at the center of a racket, the shape of the curve also remains the same, but MgR/I increases.

Thanks for posting all the good info. Here is what I got. I have tune my daughter's racquet (she is 15) for a long time using Travlerajm's method and it seem to work very well. I ended up with 324g with a balance of 32.7 and swingweight of 310. She was using a windsheild style forehand and it seemed that the shots where very spinny but not a lot of power. I tried all kinds of strings and tensions and I could not get any more power out of it. So we changed here swinging style to a Djokovic style swing which took about a month to do. Now that racquet will not work at all. The balls fly all over the place. No control at all. Tried more strings and no luck. So I tried another racquet. It was an nTour Wilson and it has some weird specs so it could be fake I don't know. Well she started hit a little better. So I just started adding lead to the top of the racquet and it got better and better and better. I added 18 grams at 12. So now her shots are controlled with a lot of spin and power. The specs are 314g with a balance of 34 and 337 swingweight. Not really close to the other specs. So I was just trying to figure out what is so different between the two styles of hitting and the curve on the racquet map. Here is a link for some WTA specs.http://www.hdtennis.com/grs/pro_racquet_specs/201207mercury-insurance-open.html
Look at Sloane Stevens racquet. Thanks again.

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I think that the relatively low static weight in combination with the relatively high sw gives a longer dwelltime, a whippier feel and thus more spin.

My guess is that with her old racket the rackethead was feeling light to her in relation to the static mass and balance. Resulting in high rackethead speed and thus easier spin, but less plow. Djokovic drives thru the ball flatter which would be harder to do when you don't feel enough weight in the rackethead.

What puzzles me is that the lighter Travlerajm racket has virtually the same MgR/I and the same shape of the curve. But if you add weight at the center of a racket, the shape of the curve also remains the same, but MgR/I increases.

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I don't think it is strange that they behave differently since they describe different things. The curve describes a swing, admittedly a very simplified and crude swing, but a least a swing where you apply a force to the racquet. MgR/I describes what happens when you attach the racquet to a cuckoo clock (exaggerating a little .

You can find other placements of the weight that might change the curve, but not MgR/I. For example you can add 10 kg at the but end or at a point around 48 cm without changing MgR/I.

I don't think it is strange that they behave differently since they describe different things. The curve describes a swing, admittedly a very simplified and crude swing, but a least a swing where you apply a force to the racquet. MgR/I describes what happens when you attach the racquet to a cuckoo clock.

You can find other placements of the weight that might change the curve, but not MgR/I. For example you can add 10 kg at the but end or at a point around 48 cm without changing MgR/I.

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That is probably why MgR/I isn't a personal ideal number, but differs with different weightdistributions.

You can see that both the swing weight (sw) and the balance (c) are in the numerator. So increasing either will increase me and make the curve steeper, but in a slightly different way.

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Tonight I played with 2 identical rackets and customized to same endspecs, 327 g, 33.3 balance, 335 sw.

One racket had all the added weight at the sides and the other one in a more polarized manner.

The played very different despite having the same curve (and the same MgR/I). I am not only referring to ball trajectory, but also the felt racketheadcontrol. The polarized one has a whippier feel than the other one.

The equivalent mass is the same for both rackets, but do you know why they play so different? I think I know why just by experience, but it would be great if there was a scientific explanation.

Tonight I played with 2 identical rackets and customized to same endspecs, 327 g, 33.3 balance, 335 sw.

One racket had all the added weight at the sides and the other one in a more polarized manner.

The played very different despite having the same curve (and the same MgR/I). I am not only referring to ball trajectory, but also the felt racketheadcontrol. The polarized one has a whippier feel than the other one.

The equivalent mass is the same for both rackets, but do you know why they play so different? I think I know why just by experience, but it would be great if there was a scientific explanation.

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You measure SW in the plane perpendicular to the stringbed, but your swing is not in that plane.

The higher twistweight frame has higher SW in the plane of your actual swing, so effective MgR/I is lower for that frame.

Could you elaborate on what you mean by that? Perhaps a drawing would help.....

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The two racquets have the same swingweight when measured in the plane normal to the stringbed, but if you mounted them in an RDC machine with the handle rotated 90 degrees to measure the swingweight in the plane of the stringbed, the racquet with higher twistweight would measure higher swingweight than the other because the added weight on the sides of the hoop is farther from the pivot point than if the weight were placed in the center of the stringbed.

Your actual swing occurs in a plane that is neither completely normal nor completely parallel to the stringbed, so the twistweight contributes to the "effective" swingweight, which is always slightly higher than the measured swingweight. In most cases, the difference between measured swingweight and "effective" swingweight is consistent enough that it is not worth worrying about the difference. But the example presented (with 2 frames of equal mass, balance, and measured SW, but different mass distributions) is a case where the difference does come into play.

I have to go catch a plane this morning, so I'll leave a drawing to someone else.